Subgroups — #21432840

Order 7552.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273734	C₂	1	2	yes	maximal, complemented
2	#16837480	C₂	1	2	yes	maximal, complemented
3	#16837474	C₂	1	4	yes	maximal, complemented
4	D₄×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837369	C₂	1	2	yes	maximal, complemented
7	C₄×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
9	#14273734	—	2	8	no	—
10	C₂×#11745664	C₂²	1	4	yes	—
11	C₂×#11745664	C₂²	1	1	yes	characteristic
12	C₂×#11745664	C₂²	1	1	yes	characteristic
13	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
19	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
20	#14273638	C₂²	1	2	yes	—
21	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
22	C₂×#41×C₅₉	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
24	#127×C₅₉	C₄	1	1	yes	characteristic
25	#124×C₅₉	C₄	1	2	yes	—
26	#121×C₅₉	C₄	1	2	yes	—
27	#121×C₅₉	C₄	1	2	yes	—
28	C₄×D₄×C₅₉	C₂²	1	2	yes	—
29	C₄×D₄×C₅₉	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	#41×C₅₉	—	2	4	no	—
32	#41×C₅₉	—	2	4	no	—
33	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₅₉	C₂×C₄	1	4	yes	—
36	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
44	C₂²⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
45	C₂²⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
46	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂³	1	1	yes	characteristic
49	C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	#11745664	—	2	2	no	—
51	#11745664	—	2	8	no	—
52	#11745664	D₄	1	4	yes	complemented
53	C₂×C₅₉⋊C₈	—	2	4	no	—
54	C₂×C₅₉⋊C₈	—	2	8	no	—
55	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
56	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
57	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
58	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
59	C₂³×C₅₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₅₉	—	2	4	no	—
62	D₄×C₅₉	—	2	8	no	—
63	D₄×C₅₉	—	2	4	no	—
64	D₄×C₅₉	—	2	4	no	—
65	D₄×C₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	—	2	4	no	—
69	C₂×C₄×C₅₉	—	4	1	no	—
70	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₅₉	#41	1	2	yes	—
78	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₅₉⋊C₈	—	2	2	no	—
80	C₅₉⋊C₈	—	2	2	no	—
81	C₅₉⋊C₈	—	2	2	no	—
82	C₅₉⋊C₈	C₂×D₄	1	4	yes	—
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	—	2	4	no	—
86	C₂²×C₅₉	—	2	4	no	—
87	C₂²×C₅₉	—	2	4	no	—
88	C₂²×C₅₉	C₄×D₄	1	2	yes	—
89	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₅₉	—	2	4	no	—
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	C₄×D₄	1	2	yes	—
95	C₄×C₅₉	#131	1	2	yes	complemented
96	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
98	#2935	—	59	1	no	maximal, Sylow
99	C₂×C₅₉	—	2	2	no	—
100	C₂×C₅₉	—	4	4	no	—
101	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
103	C₂×C₅₉	#568	1	1	yes	characteristic
104	C₂×#131	—	59	2	no	—
105	#532	D₅₉	1	1	yes	characteristic
106	#436	—	59	2	no	—
107	#435	—	59	4	no	—
108	C₈×D₄	—	59	2	no	—
109	C₂×C₄⋊C₈	—	59	1	no	—
110	#408	—	59	2	no	—
111	C₄×M₁₆	—	59	1	no	—
112	C₅₉	#2935	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	C₂×D₅₉	1	2	yes	—
114	#131	—	118	8	no	—
115	C₂×M₁₆	—	59	1	no	—
116	C₂×M₁₆	—	59	4	no	—
117	C₂×M₁₆	—	59	1	no	—
118	C₂²×C₈	—	59	2	no	—
119	C₂²×C₈	—	59	4	no	—
120	C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
121	#127	Dic₅₉	1	1	yes	characteristic
122	#124	Dic₅₉	1	2	yes	—
123	#121	Dic₅₉	1	2	yes	—
124	#121	Dic₅₉	1	2	yes	—
125	C₄×D₄	C₂×D₅₉	1	2	yes	—
126	C₄×D₄	C₂×D₅₉	1	2	yes	—
127	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
128	C₄⋊C₈	—	59	1	no	—
129	C₄⋊C₈	—	59	2	no	—
130	C₄⋊C₈	—	59	1	no	—
131	C₂²⋊C₈	—	59	4	no	—
132	C₂²⋊C₈	—	59	4	no	—
133	#97	—	59	2	no	—
134	C₄×C₈	—	59	2	no	—
135	#41	—	2	4	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
138	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
139	C₂×D₄	C₂×Dic₅₉	1	4	yes	—
140	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
141	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
142	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
145	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
146	M₁₆	—	59	4	no	—
147	M₁₆	—	118	8	no	—
148	M₁₆	—	118	2	no	—
149	C₂×C₈	—	59	2	no	—
150	C₂×C₈	—	59	2	no	—
151	C₂×C₈	—	59	2	no	—
152	C₂×C₈	—	59	2	no	—
153	C₂×C₈	—	118	8	no	—
154	C₂×C₈	—	118	4	no	—
155	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₅₉	1	4	yes	—
158	C₂²⋊C₄	C₂×Dic₅₉	1	4	yes	—
159	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
160	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
161	C₄²	C₂²×D₅₉	1	1	yes	characteristic
162	C₄²	C₂²×D₅₉	1	1	yes	characteristic
163	C₂³	C₂²×Dic₅₉	1	4	yes	—
164	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	D₄×D₅₉	1	2	yes	—
175	C₂×C₄	#11745687	1	2	yes	—
176	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₅₉	1	4	yes	—
180	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
182	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
183	C₈	—	59	4	no	—
184	C₈	—	118	2	no	—
185	C₈	—	118	2	no	—
186	C₈	—	118	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	D₄×Dic₅₉	1	2	yes	—
193	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	4	no	—
198	C₄	D₄×Dic₅₉	1	2	yes	—
199	C₄	#14273734	1	2	yes	complemented
200	C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
201	C₄	C₂×#11745687	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×D₄×Dic₅₉	1	1	yes	characteristic
205	C₂	C₂×#14273734	1	1	yes	characteristic
206	C₂	#16838169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#532	C₂
hypercenter	#532	D₅₉
hypocenter	C₅₉	#2935
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#14273734	C₂	semidirect
2. #16837480	C₂	semidirect
3. #16837474	C₂	semidirect
4. D₄×C₅₉⋊C₈	C₂	semidirect
5. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
6. #16837369	C₂	semidirect
7. C₄×#11745664	C₂	semidirect
52. #11745664	D₄	semidirect
95. C₄×C₅₉	#131	semidirect
112. C₅₉	#2935	semidirect
199. C₄	#14273734	semidirect